LOCAL COHOMOLOGY: COMBINATORICS AND D-MODULES
A Thesis by ROBERTO BARRERA III
Submitted to the O ce of Graduate and Professional Studies of Texas A&M University in partial ful llment of the requirements for the degree of DOCTOR OF PHILOSOPHY
Chair of Committee, Laura Matusevich Committee Members, Frank Sottile Catherine Yan John Keyser Head of Department, Emil Straube
August 2017
Major Subject: Mathematics
Copyright 2017 Roberto Barrera III ABSTRACT
In this dissertation, we study combinatorial and D-module theoretic aspects of local cohomology.
Viewing local cohomology from the point of view of A-hypergeometric systems, the quasidegree set of the non-top local cohomology modules supported at the max- imal ideal m of a toric ideal determine parameters β where the rank of the corre- sponding hypergeometric system is higher than expected. We discuss the Macaulay2 package, Quasidegrees, and its main functions. The main purpose of Quasidegrees is to compute where the rank of the solution space of an A-hypergeometric system is higher than expected. Local duality gives a vector space isomorphism between local cohomology and Ext-modules. However, the proof of local duality is nonconstructive. We recall a combinatorial construction by Irena Peeva and Bernd Sturmfels to minimally resolve codimension 2 lattice ideals. With motivation coming from A-hypergeometric sys- tems, we use the construction by Peeva and Sturmfels to construct an explicit local duality isomorphism for codimension 2 lattice ideals. In general, local cohomology modules of a ring may not be nitely generated. However, they still may possess other niteness properties. In 1990, Craig Huneke asked if the number of associated prime ideals of a local cohomology module is - nite. Using characteristic free D-module techniques inspired by Glennady Lyubeznik, we answer Huneke's question in the a rmative for local cohomology modules over Stanley-Reisner rings.
ii DEDICATION
This dissertation is dedicated to my family for their continued love and support in this endeavor.
iii ACKNOWLEDGMENTS
I am deeply thankful for the guidance, support, opportunities, and seemingly endless patience from my advisor Dr. Laura Matusevich during my years of graduate study at Texas A&M University. I would also like to thank my research committee Dr. Frank Sottile, Dr. Catherine Yan, and Dr. John Keyser along with Dr. Maurice Rojas and Dr. Anne Shiu for their comments and suggestion in my work. I am also grateful to Dr. Daniela Ferrero for giving me my rst taste of research in mathematics. Finally, I am indebted to my friends, family, professors, and sta in the Mathe- matics Department at Texas A&M University for their continuous support, encour- agement, advice, and help that as allowed for the completion of this dissertation.
iv CONTRIBUTORS AND FUNDING SOURCES
Contributors
This work was supported by a dissertation committee consisting of Professor Laura Matusevich [advisor], Professor Frank Sottile, and Professor Catherine Yan of the Department of Mathematics and Professor John Keyser of the Department of Computer Science and Engineering. All of the work in chapter 5 is joint work with Jerey Madsen of Purdue Univer- sity and Ashley Wheeler of University of Arkansas under the guidance of Wenliang Zhang of University of Illinois at Chicago. The work in chapter 5 was done during a Mathematics Research Communities held by the American Mathematical Society in 2015. All other work conducted for the dissertation was completed by the student in- dependently.
Funding Sources
Graduate study was supported by a Teaching Assistantship from Texas A&M University.
v TABLE OF CONTENTS
Page
ABSTRACT ...... ii
DEDICATION ...... iii
ACKNOWLEDGMENTS ...... iv
CONTRIBUTORS AND FUNDING SOURCES ...... v
TABLE OF CONTENTS ...... vi
LIST OF FIGURES ...... viii
1. INTRODUCTION ...... 1
2. PRELIMINARIES ...... 4
2.1 Commutative algebra ...... 4 2.2 Homological algebra ...... 7 2.3 Local cohomology ...... 8
3. COMPUTING QUASIDEGREES AND A-HYPERGEOMETRIC SYSTEMS 11 3.1 Combinatorics of quasidegree sets ...... 12 3.2 Quasidegrees ...... 14 3.3 Quasidegrees and hypergeometric systems ...... 16
4. AN ISOMORPHISM OF LOCAL DUALITY FOR CODIMENSION 2 TORIC IDEALS ...... 21
4.1 Introduction ...... 21 4.2 Combinatorics of resolving codimension 2 toric ideals ...... 22 4.3 Combinatorics of local cohomology for codimension 2 toric ideals . . . 27 4.4 A local duality map for codimension 2 toric ideals ...... 33 4.5 Examples ...... 38
5. ASSOCIATED PRIME IDEALS OF LOCAL COHOMOLOGY MODULES OVER STANLEY-REISNER RINGS ...... 47
vi 5.1 Simplicial complexes and Stanley-Reisner rings ...... 47 5.2 Dierential operators on Stanley-Reisner rings ...... 51 5.3 Associated primes of local cohomology modules of Stanley-Reisner rings 59
REFERENCES ...... 67
APPENDIX A. MACAULAY2 SCRIPTS FOR QUASIDEGREES ...... 69
vii LIST OF FIGURES
FIGURE Page
3.1 A general staircase diagram ...... 13
3.2 The staircase diagram for Example 3.1.1 ...... 14
4.1 Master tree ...... 25
4.2 Schematic of the dierentials in a quadrangle complex ...... 26
4.3 The vector arrangement in case 1 of Proposition 4.3.3 ...... 30
4.4 The vector arrangement in case 2 of Proposition 4.3.3 ...... 30
4.5 The vector arrangement in case 3 of Proposition 4.3.3 ...... 31
4.6 Gale diagram in the proof of Proposition 4.4.3 ...... 36
xv 4.7 Gale diagram of the complement of j ...... 37 nsupp( xu ) 4.8 conv(A) and the Gale diagram of B in Example 4.5.1 ...... 39
4.9 conv(A) and the Gale diagram of B in Example 4.5.2 ...... 41
4.10 conv(A) and the Gale diagram of B in Example 4.5.3 ...... 42
4.11 conv(A) and the Gale diagram of B in Example 4.5.4 ...... 44
4.12 The two syzygy quadrangles of IA...... 44
5.1 A simplicial complex that is not a T -space ...... 49
5.2 A pure simplicial complex that is not a T -space ...... 49
5.3 A pure T -space ...... 49
5.4 A non-pure T -space ...... 50
viii 1. INTRODUCTION
Local cohomology was introduced by Alexander Grothendieck to prove Lefschetz- type theorems. Since its introduction, local cohomology has found numerous con- nections with other areas of mathematics. Algebraically, local cohomology modules can be used to measure the dimension and depth of a module on an ideal. As a consequence, local cohomology can be used to test if a ring is Cohen-Macaulay or Gorenstein. Local cohomology can also assist in answering the question of how many generators an ideal has up to radical. The feature of local cohomology that allows us to answer this question is the fact that local cohomology of a module supported at an ideal is the same as the local cohomology of the module supported at the radical of the ideal, that is Hi(M) = H√i (M) for an I I R-module M and ideal I ⊂ R. Furthermore, if I is generated by n elements, there is no local cohomology in cohomological degree larger than n. From the D-module point of view, local cohomology is used to determine the pa- rameters at which the rank of an associated system of partial dierential equations, called the A-hypergeometric system with parameter β, is higher than expected. A- hypergeometric systems have an associated toric ideal in the polynomial ring. Toric ideals have a rich combinatorics that is used to gain an understanding of their local cohomology. Conversely, D-modules help us gain a better understanding of the struc- ture of local cohomology modules. For example, D-modules are used to determine if the number of associated prime ideals of a local cohomology module is nite.
For the geometric interpretation of local cohomology, let X be a topological space. A subset V ⊂ X is locally closed if it is the intersection of an open subset and a closed subset of X. Let F be an abelian sheaf on X, V be a locally closed in X, and
1 U is an open subset of X in which V is closed in. Set ΓV (X, F ) to be the subset of sections s ∈ Γ(U, F ) such that s restricted to the complement U \ V is zero. One can show that this de nition is independent of U and ΓV (X, −) is left exact. The i-th right derived functor of ΓV (X, −) evaluated at F is the i-th local cohomology group i . Since is left exact, 0 ∼ . HV (X, F ) ΓV (X, −) HV (X, −) = ΓV (X, −) In this dissertation, we study computational, combinatorial, and D-module as- pects of local cohomology. This dissertation is organized as follows. In Chapter 2, we give the de nition and properties of local cohomology. We also recall basic notions from commutative algebra and homological algebra that will be used throughout this dissertation. Chapter 3 begins by de ning the quasidegree set of a module. When the module is presented by a matrix with monomial entries, the quasidegree set can be computed combinatorially using standard pairs of monomial ideals coming from the entries of the presentation matrix. We discuss the main functions of the Macaulay2 package Quasidegrees. The chapter concludes with a discussion on the motivation for writing
Quasidegrees, namely A-hypergeometric systems. We de ned A-hypergeometric sys- tems and recall results by Matusevich, Miller, and Walther that give a connection between quasidegrees of the non-top local cohomology modules of toric ideals and the parameters at which the dimension of the corresponding A-hypergeometric system is higher than expected. The goal of chapter 4 is to describe an isomorphism for local duality for the non-top local cohomology modules of codimension 2 lattice ideals. The chapter begins by describing the combinatorial construction for the minimal free resolution of codimension 2 lattice ideals by Irena Peeva and Bernd Sturmfels, found in [12].
We then show combinatorial properties of the Ext-modules of codimension 2 lattice ideals. Using the combinatorial data, we construct an isomorphism for local duality
2 of codimension 2 lattice ideals. Craig Huneke asked whether the set of associated primes of a local cohomology module is nite. The goal of Chapter 5 is to answer Huneke's question for local cohomology modules of Stanley-Reisner rings. The chapter begins by recalling the necessary de nitions and properties for simplicial complexes and Stanley-Reisner rings as well as the needed background from the theory of D-modules. Using D- module techniques inspired by Gennady Lyubeznik, chapter 5 concludes by answer- ing Huneke's question question in the a rmative for Stanley-Reisner rings whose associated simplicial complex is a T-space.
3 2. PRELIMINARIES
This chapter provides the necessary de nitions and properties in commutative and homological algebra that will be used throughout this dissertation. We follow de nitions and conventions in [5].
For the remainder, let N = {0, 1, 2,...} and k be a eld of characteristic 0 unless otherwise speci ed. Let be a polynomial ring in variables S = k[x1, . . . , xn] = k[x] n over the eld k. We use multi-index notation throughout.
Notation 2.0.1. Let n. Then a denotes the monomial a = (a1, . . . , an) ∈ N x
a1 an . A polynomial in may be written as P a for some . If x1 ··· xn S a cax ca ∈ k n, then P a ± is a Laurent polynomial. a ∈ Z a cax ∈ k[x ]
2.1 Commutative algebra
Let R be a commutative ring, M be an R-module, and U ⊂ R be a multiplica- tively closed subset. The localization of M at U, written M[U −1] or U −1M, is the set of equivalence classes of pairs (m, u) with m ∈ M and u ∈ U with the equivalence relation (m, u) ∼ (m0, u0) if there is an element v ∈ U such that v(mu0 − m0u) = 0 in M. Denote the equivalence class (m, u) by m/u. The localization M[U −1] is an R-module by de ning
m/u + m0/u0 = (mu0 + m0u)/(uu0) and r(m/u) = (rm)/u for m, m0 ∈ M, u, u0 ∈ U, and r ∈ R. If M = R, then the localization is a ring with multiplication de ned as (r/u)(r0/u0) = (rr0)/(uu0). There is a natural inclusion M → M[U −1] de ned by m 7→ m/1. If U = R \ P for some prime ideal P ⊂ R, then
4 i we write the localization at P as MP . If f ∈ R and U = {f : i ≥ 0}, then we write the localization at f as Mf . A commutative ring R is graded if we can write R as a decomposition as an additive group L such that for an abelian monoid . We R = i∈G Ri RiRj ⊂ Ri+j G may call R a G-graded ring and say R is G-graded to emphasize which grading on R we are using. Similarly, an R-module M is graded by G if R is G-graded and has a decomposition L such that . An element M M = i∈G RiMj ⊂ Mi+j m ∈ M is homogeneous if m ∈ Mi for some i ∈ G and the element m is said to have degree i. It may be convenient to shift the degrees of the module. If M is a
G-graded module, de ne M(g) to be the G-graded module where M(g)i = Mg+i. Let f : M → N be a homomorphism of G-graded R-modules. Then f is said to be graded, or homogeneous, of degree d if f(Mi) ⊂ Ni+d for all i.
Example 2.1.1. Let . Then has an -grading by setting S = k[x1, . . . , xn] S N S = L S where S = ·{xa : a + ··· + a = i} is the -vector space generated by i∈N i i k 1 n k monomials of degree i. This is the standard grading on S.
Example 2.1.2. Let . Then has a n-grading by setting S = k[x1, . . . , xn] S N S = L S where S = ·{xa} is the -vector space with basis xa. a∈Nn a a k k
A proper ideal I ⊂ R is prime if xy ∈ R and x∈ / I implies y ∈ I. If M is an R-module, then a prime ideal P ⊂ R is said to be associated to M if there is an m ∈ M such that P = ann(m) := {r ∈ R : rm = 0}. We call P an associated prime ideal of M, or simply an associated prime if the context is understood. The set of associated primes of M is denoted Ass(M). A module M is called P -primary if P is the only associated prime of M. If I ⊂ R is an ideal and M = R/I, we say that P is an associated prime of I if P is an associated prime of R/I. Similarly, we say that I is P -primary when R/I is P -primary.
5 A prime ideal P is minimal over an ideal I if P contains I and does not contain any other prime ideal that contains I. The minimal primes of a module M are the minimal primes over the annihilator ann(M) := {r ∈ R : rM = 0} of M. The primes in Ass M that are not minimal are called embedded primes of M. A submodule N ⊂ M is irreducible if it cannot be written as the intersection of two distinct submodules of M. If N ⊂ M is a submodule of M, a decomposition of N is an expression of the form N = N1 ∩ · · · ∩ Nl where the Ni are submodules of M. If each Ni is irreducible, such a decomposition is called an irreducible decomposition. If each Ni is primary, the decomposition is called a primary decomposition. If we cannot omit any Ni in the decomposition, then we call such a decomposition irredundant.
Theorem 2.1.3 (Primary decomposition). Let R be a Noetherian ring and M be an R-module.
1. If N ⊂ M is an irreducible submodule, then N is primary.
2. If N = N1 ∩ N2 ∩ · · · ∩ Nl with Ass(M/Ni) = {Pi} is an irredundant primary
decomposition of a proper submodule N ⊂ M, then Ass(M/N) = {P1,...,Pl}.
3. Every proper submodule of M has a primary decomposition.
The dimension of a ring R, denoted dim(R) is the supremum of the lengths of chains of distinct prime ideals in R where the length of the chain P0 ⊂ P1 ⊂ · · · ⊂ Pr of prime ideals is r. The dimension of a module M is taken to be dim(R/ annR(M)). The height of a prime ideal P ⊂ R, denoted height(P ), is the supremum of the lengths of strictly decreasing chains of prime ideals starting P = P0 ) P1 ) ··· ) Pl from P . We remark that height(P ) = dim(RP ). The height of an ideal I ⊂ R is the minimum of the heights of primes that contain I.
6 Let R be a ring and M be an R-module. A nonzero divisor in R is an element x ∈ R such that for every r 6= 0, xr 6= 0 A sequence x1, . . . , xn of elements of R is a regular sequence if x1, . . . , xn generate a proper ideal of R and xi is a nonzero divisor in R/(x1, . . . , xi−1). There is an analogous de nition for modules. An element x ∈ R is a nonzero divisor on M, or M-regular, if xm 6= 0 for all 0 6= m ∈ M. A sequence x1, . . . , xn is an M-sequence, or an M-regular sequence, if (x1, . . . , xn)M 6= M and xi is a nonzero divisor on M/(x1, . . . , xi−1)M. An ideal that is generated by a regular sequence is called a complete intersection.
Let I be an ideal of a ring R and let M be a nitely generated R-module such that IM 6= M. The depth of I on M, denoted depth(I,M), is the length of the longest regular sequence on M contained in I. If R is local or homogeneous, then depth(M) := depth(m,M) where m is the maximal homogeneous ideal in R.
2.2 Homological algebra
Let R be a commutative ring with 1 and M be a left R-module. A sequence
φ1 φ2 F• : 0 ←− F0 ←− F1 ←− F2 ←−· · ·
of maps of R-modules is a chain complex, or simply a complex, if φi+1 ◦ φi = 0 for all i. The module Fi is said to be in homological degree i. The length of the chain complex F• is the largest homological degree l in which Fl is a nonzero module in
F•. We call a chain complex exact if ker φi = im φi+1. If the module M is graded, we require the homomorphisms φi to be graded of degree 0.
A projective resolution of M is a chain complex P• of projective R-modules
φ1 φ2 P• : 0 ←− P0 ←− P1 ←− P2 ←−· · ·
7 that is exact everywhere except in homological degree 0, where we require M =
P0/ im φ1. The projective dimension of M is the minimum length of a projective resolution of M. We often augment the projective resolution P• by placing 0 ←−
M ←− F0 at the left end to make P• exact everywhere. Dually, an injective resolution of M is a chain complex J • of injective R-modules
φ φ φ J • : 0 −→ J 0 −→0 J 1 −→1 J 2 −→·2 · · .
The injective dimension of M is the minimum length of an injective module of M.
The functor HomR(−,N) is a contravariant left-exact functor. Let M be an
1 R-module and P : P0 ← P ← · · · a projective resolution of M. The i-th ho- mology module of the complex Hom(M,N) = Hom(P,N) : 0 → HomR(P0,N) → 1 is denoted i . Alternatively, i is the - HomR(P ,N) → · · · ExtR(M,N) ExtR(M,N) i
i th right derived functor R Hom(M, −)(N). The functor HomR(M, −) is a covari- ant left-exact functor. If J : J0 → J1 → · · · is an injective resolution of N, we may also de ne i to be the -th homology of the complex ExtR(M,N) i Hom(M,N) =
• 1 Hom(M, J ) : 0 → HomR(J0,N) → HomR(J ,N) → · · · and these two notions agree.
2.3 Local cohomology
This section introduces the main algebraic object of our study. Let f = f1, . . . , fl be elements in R. The ech complex of R on f is the complex
l ˇ• M M C (f; R) : 0 −→ R −→ Rfi −→ Rfifj −→· · · −→ Rf1···fl i=1 1≤i 8 with the homomorphism being the canonical inclusions and signs coming from the Koszul resolution on f. The ech complex of an R-module M over f is the complex ˇ• ˇ• C (f; M) = C (f; R) ⊗R M. The i-th ech cohomology of f on M is Hˇ i(f; M) = Hi(Cˇ•(f; M)). Let f = f1, . . . , fl be a set of generators for the ideal I. The i-th local cohomology module of M supported at the ideal I is i ˇ i HI (M) := H (f; M). We can also de ne local cohomology of M supported at the ideal I as follows. Let t for some We call the -torsion functor. ΓI (M) = {m ∈ M : I m = 0 t ∈ N}. ΓI (−) I The I-torsion functor is left-exact. The i-th right derived functor of ΓI (M) is the -th local cohomology module supported at and denoted i , that is, i I HI (−) i i • HI (M) = H (ΓI (J ) where J • is an injective resolution of M. Viewing the local cohomology modules as the right derived functors of the I-torsion functor yields the following. Proposition 2.3.1. For any i , there exists some such that t . m ∈ HI (M) t ∈ N I m = 0 In general, local cohomology is di cult to compute. However, when M is a nitely generated module over the polynomial ring S, we have a vector space isomorphism between local cohomology and Ext-modules. 9 Theorem 2.3.2 (Local duality). Let and be the S = k[x1, . . . , xn] m = hx1, . . . , xni homogeneous maximal ideal. If M is a nitely generated graded S-module under the standard grading, then there is a graded k-vector space isomorphism i ∼ n−i Hm(M) = Ext (M,S(−n)). The following theorem tells us that the vanishing of local cohomology depends on the depth and dimension of our module. Conversely, the non-vanishing of local cohomology gives us information about the depth and dimension of the module. Theorem 2.3.3 (Vanishing of local cohomology). Let M be a nitely generated S-module. 1. If or then i . i < depth M i > dim M Hm(M) = 0 2. If or then i . i = depth M i = dim M Hm(M) 6= 0 Theorem 2.3.3 is also called depth sensitivity. The following is an immediate result of Theorem 2.3.3. Corollary 2.3.4. A nitely generated S-module M is Cohen-Macaulay if and only if M has one non-zero local cohomology module. 10 3. COMPUTING QUASIDEGREES AND A-HYPERGEOMETRIC SYSTEMS Let be a d-graded polynomial ring over a eld and let S = k[x1, . . . , xn] Z k m = hx , . . . , x i denote the homogeneous maximal ideal in S. Let M = L M 1 n β∈Zd β be a Zd-graded S-module. The true degree set of M is d tdeg(M) = {β ∈ Z | Mβ 6= 0}. The quasidegree set of M, denoted qdeg(M), is the Zariski closure in Cd of tdeg(M). In this chapter, we discuss computing the quasidegree set of a nitely generated Zd-graded S-module presented as the cokernel of a monomial matrix. We then discuss the Macaulay2 [7] package, Quasidegrees [1], whose purpose is to compute the quasidegree set of such a nitely generated Zd-graded module. Lastly, we apply Quasidegrees to compute where rank jumps of A-hypergeometric systems occur. By a monomial matrix, we mean a homogeneous map of degree zero whose matrix representation has entries that are either zero or a monomial in S. More precisely, a monomial matrix is a map ··· αj ··· . αij αi λijx . L L S(−αi) S(−αj): φ where the columns of φ are labeled by source degrees and the rows of φ are labeled by target degrees. 11 3.1 Combinatorics of quasidegree sets Suppose M is Zd-graded module that has a presentation by the monomial matrix φ : L S(−α ) → L S(−α ), that is, αj j αi i L S(−αi) M = αi . im φ The basis vector of maps to L αij . Thus the true degree set and quaside- S(−αj) i λijx gree set of M are determined by the monomials outside of the image of φ. Since φ is assumed to be a monomial matrix, the image of φ is a monomial module. We use the idea of standard pairs of monomial ideals to index the monomials outside a monomial ideal. Monomials outside of a monomial ideal are called standard monomials and are a k-basis for S/I. Given a monomial xα and a subset Z ⊂ α α β β {x1, . . . , xn}, the pair (x ,Z) indexes the monomials x · x where supp(x ) ⊂ Z.A standard pair of a monomial ideal I ⊂ R is a pair (xα,Z) satisfying: 1. supp(xα) ∩ Z = ∅, 2. all of the monomials indexed by (xα,Z) are outside of I, 3. (xα,Z) is maximal in the sense that (xα,Z) * (xβ,Y ) for any other pair (xβ,Y ) satisfying the rst two conditions. If our polynomial ring is in two or three variables, we can represent monomial ideals geometrically in the positive quadrant or orthant using a staircase diagram. Nonzero monomials correspond to their exponent vectors. We demonstrate this in two variables. Let S = k[x, y] and let I = hxα1 yβ1 , xα2 yβ2 , . . . , xαt yβt i be a monomial ideal where α1 > α2 > ··· > αt ≥ 0 and βt > βt−1 > ··· > β1 ≥ 0. The staircase diagram of I is the region of the plane that contains the exponent vectors of the 12 monomials in I. The lattice points in the nonnegative rst quadrant that are not in I represent the standard monomials. The staircase diagram of I is illustrated in Figure 3.1. (αt, βt) • ◦ • I . ◦ ◦ .. ◦ ◦ • (α1, β1) ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 α β (αi, βj) ←→ x i y j Figure 3.1: A general staircase diagram Example 3.1.1. Let with the 2-grading by letting 1 and S = k[x, y] N deg(x) = ( 0 ) 0 . Let 3 2 3 . In Figure 3.2, we represent the ideal geomet- deg(y) = ( 1 ) I = hy , xy , x yi I rically by its staircase diagram. 13 I • ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 Figure 3.2: The staircase diagram for Example 3.1.1 The standard pairs are (1, {x}), (y, ∅), (xy, ∅), (x2y, ∅), and (y2, ∅). The quasidegrees can be read from the standard pairs. In this example, the quaside- gree set is qdeg(S/I) = {(ξ, 0) : ξ ∈ C} ∪ (0, 1) ∪ (1, 1) ∪ (2, 1) ∪ (0, 2). The following routine is implemented in Quasidegrees to compute the quasidegree set of M. We rst nd a monomial presentation of M so that M is the cokernel of a monomial matrix φ. We then compute the standard pairs of the ideals generated by the rows of φ and to each standard pair we associate the degrees of the corresponding variables. 3.2 Quasidegrees In this section, we discuss some of the functions in the Macaulay2 package, Quasidegrees. The main function of Quasidegrees is the method quasidegrees. The input of quasidegrees is either a module M that can be presented as the cokernel of a monomial matrix or an ideal and the output is the quasidegree set of the mod- 14 ule. If the input is an ideal I with base ring S, then quasidegrees computes the quasidegree set of the S-module M = S/I. Quasidegrees represents the quasidegree set as a list of pairs (α, Z) with α ∈ Qd and Z ⊂ Qd where the pair (α, Z) represents the plane X α + C · β. β∈Z The union of all planes over all such pairs in the output is the quasidegree set of M. The following is an example of Quasidegrees computing the quasidegree set of an S-module with the standard grading: i1 : S=QQ[x,y,z] o1 = S o1 : PolynomialRing i2 : I=ideal(x*y,y*z) o2 = ideal (x*y, y*z) o2 : Ideal of S i3 : M=S^1/I o3 = cokernel | xy yz | 1 o3 : S-module, quotient of S i4 : Q = quasidegrees M o4 = {{0, {| 1 |}}, {0, {| 1 |, | 1 |}}} o4 : List The above example displays a caveat of Quasidegrees in that there may be some redundancies in the output. By a redundancy, we mean when one plane in the output 15 is contained in another. The redundancy above is clear: qdeg(Q[x, y, z]/hxy, yzi) = C = {ξ1 + ξ2 ∈ C | ξ1, ξ2 ∈ C}. The function removeRedundancy gets rid of redundancies in the list of planes: i5 : removeRedundancy Q o5 = {{0, {| 1 |, | 1 |}}} o5 : List 3.3 Quasidegrees and hypergeometric systems The motivation for writing the package Quasidegrees is to study the rank jumps of A-hypergeometric systems of partial dierential equations. Let A = [a1 a2 ··· an] be an integer (d × n)-matrix such that ZA = Zd, that is, the columns of A span Zd as a lattice. We also assume that the cone over the columns of A is pointed. There is a natural d-grading on by the columns of by letting Z S = k[x1, . . . , xn] A deg(xj) = aj, the j-th column of A. A module that is homogeneous with respect to this grading is said to be A-graded and have an A-grading. By the assumptions on A, S is positively graded by A, that is, the only polynomials of degree 0 are the constants. Given a matrix A and a polynomial ring S in n variables, the method 1 1 1 1 1 toAgradedRing makes S an A-graded module. For example, let A = 0 0 1 1 0 . 0 1 1 0 −2 We make the polynomial ring -graded : Q[x1, x2, x3, x4, x5] A i6 : A=matrix{{1,1,1,1,1},{0,0,1,1,0},{0,1,1,0,-2}} o6 = | 1 1 1 1 1 | | 0 0 1 1 0 | | 0 1 1 0 -2 | 3 5 16 o6 : Matrix ZZ <--- ZZ i7 : S=QQ[x_1..x_5] o7 = S o7 : PolynomialRing i8 : S=toAgradedRing(A,S) o8 = S o8 : PolynomialRing i9 : describe S o9 = QQ[x , x , x , x , x , Degrees => {{1}, {1}, {1}, {1}, 1 2 3 4 5 {0} {0} {1} {1} {0} {1} {1} {0} ------{1 }}, Heft => {1, 2:0}, MonomialOrder => {0 } {-2} ------{MonomialSize => 32}, DegreeRank => 3] {GRevLex => {5:1} } {Position => Up } The toric ideal of A in S is the binomial ideal u v n IA = hx − x : Au = Av, u, v ∈ Z i. The method toricIdeal computes the toric ideal of A in the ring S. We continue with the A and S from the above example and compute the toric ideal IA in S: 17 i10 : I=toricIdeal(A,S) 2 2 2 3 o10 = ideal (x x - x x , x x - x x , x x - x x x , x - 13 24 14 35 14 235 1 ------2 x x ) 2 5 o10 : Ideal of S We now introduce A-hypergeometric systems. Given a matrix A ∈ Zd×n as above and a parameter β ∈ Cd, the A-hypergeometric system with parameter β ∈ Cd [13], denoted HA(β), is the system of partial dierential equations: ∂|v| ∂|u| φ(x) = φ(x) for all u, v, Au = Av ∂xv ∂xu n X ∂ a x φ(x) = β φ(x), for i = 1, . . . , d. ij j ∂x i j=1 j Such systems are sometimes called GKZ-hypergeometric systems. The function gkz in the Macaulay2 package Dmodules computes this system as an ideal in the Weyl algebra. The holonomic rank of HA(β) is germs of holomorphic solutions of HA(β) rank(HA(β)) = dimC . near a generic nonsingular point In general, the rank of a hypergeometric system is an upper semi-continuous 18 function of β [10, Theorem 2.6]. Set vol(A) as d! times the Euclidean volume of conv(A∪{0}) the convex hull of the columns of A and the origin in Rd. The following theorem gives the parameters β for which rank(HA(β)) is higher than expected: Theorem 3.3.1. [10] Let HA(β) be an A-hypergeometric system with parameter β. If Ld−1 i then . Otherwise, β ∈ qdeg( i=0 Hm(S/IA)) rank(HA(β)) > vol(A) rank(HA(β)) = vol(A). Since Theorem 3.3.1 was the initial motivation for Quasidegrees, the package has a method quasidegreesLocalCohomology (abbreviated QLC) to compute the quasidegree set of the local cohomology modules i . If the input is an integer Hm(S/IA) and the -module , then the method computes i . If the input i S S/IA qdeg(Hm(S/IA)) is only the module S/IA, the method computes the quasidegree set in Theorem 3.3.1. We use graded local duality to compute the local cohomology modules supported at m of a nitely generated A-graded S-module. The algorithm implemented for QLC is essentially the algorithm for quasidegrees applied to the Ext-modules of M with the additional twist of εA coming from local duality. For our purposes, we exploit the fact that the higher syzygies of S/IA are generated by vectors whose entires are monomials or 0 (see [11], Chapter 9). Continuing our running example, we use quasidegreesLocalCohomology to com- pute the quasidegree set of Ld−1 i : i=0 Hm(S/IA) i11 : M=S^1/I o11 = cokernel | x_1x_3-x_2x_4 x_1x_4^2-x_3^2x_5 x_1^2x_4-x_2x_3x_5 x_1^3-x_2^2x_5 | 1 o11 : S-module, quotient of S i12 : quasidegreesLocalCohomology M 19 o12 = {{| 0 |, {| 1 |}}} | 0 | | 0 | | 1 | | -2 | o12 : List Thus d−1 ! M i h 0 i h 1 i qdeg H (S/IA) = 0 + · 0 . (3.1) m 1 C −2 i=0 To con rm our computations, we use the methods gkz and holonomicRank from the package Dmodules to compute rank(HA(0)) and rank(HA(β)) for two dierent choices of β in (3.1) and demonstrate a rank jump: i13 : holonomicRank gkz(A,{0,0,0}) -- vol A in this case o13 = 4 i14 : holonomicRank gkz(A,{0,0,1}) o14 = 5 i15 : holonomicRank gkz(A,{3/2,0,-2}) o15 = 5 The source code for quasidegrees and quasidegreeslocalcohomology can be found in the appendix. 20 4. AN ISOMORPHISM OF LOCAL DUALITY FOR CODIMENSION 2 TORIC IDEALS 4.1 Introduction Let be the polynomial ring over in variables and let S = k[x1, . . . , xn] k n A = d×n be an integer matrix of full rank. We denote the -th column of by (ai,j) ∈ Z j A aj. Assume that the rst row of A is all 1's and the cone over the columns of A is pointed. The toric ideal of A in S is the binomial ideal u v IA := hx − x ∈ S : Au = Avi ⊂ S. The codimension of IA, denoted codim(IA), is n − d. For the remainder of this chapter, we assume that . Then the integer kernel of is a codim IA = 2 A kerZ(A) 2-dimensional sublattice of Zn. Let B ∈ Zn×2 be an integer matrix whose columns generate and let be the -th row of . The vector arrangement kerZ(A) bi = (bi1 bi2) i B of the rows of B in the Euclidean plane is called the Gale diagram of A, or of IA. The Gale diagram of is unique up to an action on 2. Gale diagrams of A SL2(Z) Z codimension 2 toric ideals characterize Cohen-Macaulayness [12]. Theorem 4.1.1. A toric ideal IA is not Cohen-Macaulay if and only if A has a Gale diagram that intersects each of the four open quadrants of Z2. We do not consider Cohen-Macaulay codimension 2 toric ideals because their local cohomology is well understood. By interchanging the rows of B and corresponding columns of A, we may assume for the remainder that the rst four columns of B 21 have the following signs: + + − + − − B = + − . . . . . Notation 4.1.2. The support of a vector d is the set . b ∈ Z supp(b) := {i : bi 6= 0} A vector d can be written as where the entries of and are b ∈ Z b = b+ − b− b+ b− non-negative and . If is a matrix, write to be the supp(b+) ∩ supp(b−) = ∅ B Bi u u i-th column of B. The positive support of a monomial x is supp(x ) = {xi : ui > 0} u u and the negative support of a monomial x to be nsupp(x ) = {xi : ui < 0}. Since we assume the codimension of IA is 2, the vanishing of local cohomology means that the only non-top local cohomology occurs in cohomological degree n − 3. By local duality, we have the following isomorphism. Theorem 4.1.3 (Graded local duality). There is a vector space isomorphism 3 ∼ n−3 ExtS(S/IA,S)α = Homk(Hm (S/IA)−α+A , k) where m = hx1, . . . , xni and A is the sum of the columns of A. The goal of this chapter is to produce an explicit isomorphism for Theorem 4.1.3. 4.2 Combinatorics of resolving codimension 2 toric ideals Irena Peeva and Bernd Sturmfels give a combinatorial construction for the min- imal free resolution of a codimension 2 toric ideal in [12]. In this section, we recall their construction. ∗ Let GA be the Gale diagram of A. Rotating the GA by π/2 gives the Gale diagram ∗ ∗ ∗ where ∗ . After cyclically relabeling the GA = {b1,..., bn} bi = (−bi,2, bi,1) vectors ∗, we may assume that no element of ∗ lies in the interior of a cone bi GA . Let be the minimal generating set of pos(bi, bi+1) = {λbi + µbi+1 : λ, µ ∈ R≥0} Hi 22 the semigroup 2 ∗ ∗ . The Hilbert basis of ∗ is de ned to be the set Z ∩ pos(bi , bi+1) GA 2 HA = {u ∈ Z : u, −u ∈ H1 ∪ · · · ∪ Hn}, where the antipodal points of HA are identi ed. The elements of the Hilbert basis HA correspond to the generators of the lattice ideal IA. Theorem 4.2.1. If IA is not a complete intersection, the elements u ∈ HA corre- (Bu)+ (Bu)− spond to a unique set of A graded binomial generators x − x of IA. A lattice polytope is the convex hull of a nite set of integer lattice points. A lattice polytope is said to be primitive if has no lattice points in its interior. A primitive parallelogram is a syzygy quadrangle if and only if each vertex of the parallelogram is supported by at least one vector in the Gale diagram GA, that is, if v is a vertex of the parallelogram, then there is some bi such that bi · v ≥ bi · p for every point p in the parallelogram. Furthermore, a primitive parallelogram is a syzygy quadrangle if and only if its diagonal represents a minimal generator of IA. Let the generators of IA corresponding to diagonals of a syzygy quadrangle be α = xu+ xtxp − xu− xsxr and β = xv+ xsxp − xv− xtxr where xp is the greatest common factor between the rst term of α and the rst term of β, xt is the greatest common factor between the rst term of α and the second term of β, xs is the greatest common factor between the second term of α and the rst term of β, and xr is the greatest common factor between the second term of α and the second term of β. The edges of the syzygy quadrangle are represented by 23 the binomials γ = xu+ xv+ x2p − xu− xv− x2r and δ = xu+ xv− x2t − xu− xv+ x2s. There is a chain complex associated to the syzygy quadrangle Pu called the quad- rangle complex of Pu v p v r v t v s −xs x + x x − x −x − x −x + x u r u p u s u t xt x − x x + x x − x x + x xr −xt −xs 0 0 p p r ( α β γ δ ) 0 −→ S −−−−−→−x S4 −−−−−−−−−−−−−−−−−−−−−→0 0 x x S4 −−−−−−→ S. (4.1) The twists of S4 in homological degree one are S(−A · (u+ + t + p)) ⊕ S(−A · (v+ + s + p))⊕ S(−A · (u+ + v+ + 2p)) ⊕ S(−A · (u+ + v− + 2t)), the twists of S4 in homological degree two are S(−A · (u+ + t + 2p + v+)) ⊕ S(−A · (v+ + s + p + v− + r))⊕ S(−A · (u+ + v+ + 2p + v− + t)) ⊕ S(−A · (u+ + v− + 2t + v+ + s)), and the twist of S in homological degree three is S(−A · (u+ + t + 2p + v+ + s)) = S(−A · (B1+ + B2+)). We now present the combinatorial construction found in [12] for the minimal free resolution of . For 2, denote the convex hull of , , , and by IA u, v ∈ Z (0, 0) u v u + v) [u, v] = conv((0, 0), u, v, u + v). 24 We construct a directed tree of primitive quadrangles as follows. Let the unit square [(0, 1), (1, 0)] be the root of this tree. For a quadrangle that is not the unit square, there are two outgoing edges to quadrangles [u + v, v] and [u, u + v]. Since the unit square has two representations, [(0, 1), (1, 0)] = [(0, 1), (−1, 0)], the root has four outgoing edges. The resulting tree is called the master tree and is depicted in Figure 4.1, as found in [12]. Figure 4.1: Master tree The subgraph of the master tree consisting of all syzygy quadrangles is called the homology tree of IA, denoted TA. Moving along the tree corresponds to applying the generators of SL(2, Z) and their inverses to the parallelograms where the generators of are 1 0 and 1 1 . SL(2, Z) s = ( 1 1 ) t = ( 0 1 ) We now construct the quadrangle complex combinatorially for the unit square. Syzygy quadrangles corresponds to the minimal third syzygies. The minimal second syzygies are the triangles, called syzygy triangles, obtained by deleting a vertex from a syzygy quadrangle. The generators of IA are the edges and diagonals of the syzygy quadrangle obtained by deleting a vertex from the syzygy triangles. The quadrangle 25 complex for a syzygy quadrangle is F : 0 → S → S4 → S4 → S. The dierentials are given schematically in Figure 4.2. • • • • • • • • 7→ − − + + • • • • • • • • • • • 7→ − + + • • • • • • • • 7→ − • • Figure 4.2: Schematic of the dierentials in a quadrangle complex The algebraic information for the quadrangle complex F is found as follows. Let m3 = B1+ + B2+, m4 = m3 − B1. m2 = m3 − B2, m1 = m3 − B1 − B2. Label a vertex of the unit square by the monomial with exponent vector being the corresponding supporting vector. Note that gcd(xm1 , xm2 , xm3 , xm4 ) = 1. Let D be an ordered subset of the ordered monomials [xm1 , xm2 , xm3 , xm4 ]. Then D is a generator that corresponds to a syzygy quadrangle, triangle, edge, or vertex so 26 that the quadrangle complex for the unit square is F : S → S4 → S4 → S has basis elements being the symbols [D]. The dierential of the quadrangle complex is de ned to be X D 7→ sign(xm,D) · gcd(D \ xm) · [D \ xm] xm∈D where sign(xm,D) = (−1)t+1 if xm is the t-th monomial in the ordered list [D]. If the syzygy quadrangle P is not the unit square, there is an element u ∈ SL(2, Z) that acts on the unit square to give P . To get the syzygy quadrangle complex for P , let m3 = (B · u)1+ + (B · u)2+ and then proceed as before. The minimal free resolution of IA is M FIA = FP P ∈TA where the sum runs over the homology tree of IA. 4.3 Combinatorics of local cohomology for codimension 2 toric ideals Without loss of generality we assume that S/IA has exactly one syzygy quadran- gle. From Theorem 4.1.3, we are interested in 3 . Applying ExtS(S/IA,S) Hom(−,S) to the syzygy complex (4.1) and taking homology gives 3 ∼ s t r p Ext (S/IA,S) = S(A · (B1+ + B2+))/hx , x , x , x i. Thus the standard monomials of s t r p are a -basis for 3 hx , x , x , x i k ExtS(S/IA,S) and are indexed by the standard pairs of hxs, xt, xr, xpi. We will see in this section that if (xu,Z) is a standard pair of hxs, xt, xr, xpi, then Z is a codimension 2 face of conv(a1,..., an), the convex hull of the columns of A. We will use these codimension 2 faces to make a map between the vector spaces in local duality. The local cohomology supported at the maximal ideal of a toric ring can be com- 27 puted using the Ishida complex. Let P = conv(A) be the convex hull of the columns of A and label the vertices of P by the corresponding indeterminates x1, . . . , xn. We allow xi to possibly lie in the interior of a face. Note that dim(P ) = d − 1 since the rst row of A is assumed to be all 1's. Denote the set of t-dimensional faces of by t . In general, if the dimension of the largest linear subspace of has P FP pos(A) dimension l, then P is taken to be a codimension l + 1 transverse linear section of pos(A). The largest linear subspace of a polyhedron P is the lineality space of P . The Ishida complex of IA (or of A) is the cocomplex •Ω M M A : S/IA → (S/IA)v → · · · → (S/IA)F → (S/IA)x1···xn . v∈ 0 d−2 FP F ∈FP The dierentials consist of natural localization maps ∂ (S/IA)F ∈F t−1 → (S/IA)G∈F t with signs as in the algebraic cochain complex of P [4, Theorem6.2.5]. The terms and are in cohomological degree 0 and respectively. We remark S/IA (S/IA)x1,...,xn d that the Ishida complex is a subcomplex of the ech complex. Theorem 4.3.1. Let m be the multigraded maximal ideal of S/IA and M be an S/IA-module. Then i ∼ i •Ω Hm(M) = H (M ⊗ A). For our purposes, we are interested in n−3 . By Theorem 4.3.1, this is the Hm (S/IA) cohomology of the sequence M M (S/IA)F → (S/IA)F → (S/IA)x1···xn . d−3 d−2 F ∈FP F ∈FP We now analyze the combinatorics of the polytope P that will allow us to compute local cohomology in later sections. For the remainder, let xp, xr, xs, xt be as in 28 equation 4.1. Let P be a convex set in Rd. The relative interior of P is the interior of P in the smallest a ne space that contains P . We will frequently rely on the following theorem. Theorem 4.3.2. [17, Theorem 5.6] Let P = conv(a1,..., an) and let B be as above. Then F is a face of P if and only if either F = P or 0 ∈ relint(conv(bk : xk ∈/ F )). Proposition 4.3.3. Let (xu,Z) be a standard pair of hxs, xt, xr, xpi. Then Z is a codimension 2 face of P . p t Proof. Note that Z = {x1, . . . , xn}\{xi, xj, xk, xl} where i ∈ supp(x ), j ∈ supp(x ), r s k ∈ supp(x ), and l ∈ supp(x ). Then the corresponding row vectors of B, bi, bj, , and , intersect each open quadrant of 2. Thus is a face of the polytope bk bl R Z P since 0 ∈ relint(conv{bi, bj, bk, bl}). To show Z is a codimension 2 face of P , we use Theorem 4.3.2 to construct part of the face lattice of P . There are three cases to consider for the arrangement of bi, bj, bk, bl. Case 1. Suppose {bi, bk} and {bk, bl} are two sets of linearly dependent vectors. Figure 4.3 is an example of such a vector con guration. Then removing any one vector gives a line segment through the origin and another vector emanating from the origin. In such a vector arrangement, 0 ∈/ relint(conv{bi, bj, bk}). Removing two linearly dependent vectors, say bj and bl, gives a line segment through 0 and clearly 0 is in the relative interior of such a vector arrangement. Thus, Z ∪ {xj, xl} is a face of P . Finally, removing either of bi or bk leaves a vector, which has empty interior in the Euclidean topology. We conclude that Z ∪ {xj, xl} is a codimension 1 face of P and Z is a codimension 2 face of P . Case 2. Suppose two vectors, say bi and bk, are on a line through the origin and the other two vectors, bj and bl, are not on a line through the origin. Figure 4.4 is 29 bj bi bk bl Figure 4.3: The vector arrangement in case 1 of Proposition 4.3.3 an example of such a vector con guration. Then removing either bj or bl from the con guration leaves a line segment through the origin and a vector emanating from the origin. The relative interior of the convex hull of such a con guration does not contain 0. If we remove bj and bl from the con guration, then conv{bi, bk} is a line segment through the origin and so 0 ∈ relint(conv{bi, bk}). Thus Z ∪ {xj, xl} is a codimension 1 face of P and Z is a codimension 2 face of P . bj bi bl bk Figure 4.4: The vector arrangement in case 2 of Proposition 4.3.3 Case 3. Suppose that no two of bi, bj, bk, bl lie on a line through the origin and let α, β, γ, and δ be the angles between bi and bj, bj and bk, and so on respectively. Figure 4.5 is an example of such a vector con guration. Then α+β+γ+δ = 2π. If the sum of any two consecutive angles is greater than π, then 0 is in the relative interior 30 of the convex hull of the corresponding vector arrangement. Suppose α + β < π and β + γ < π. Then γ + δ > π and 0 ∈ relint(conv{bk, bl, bi}). If we remove any two of bi, bj, bk, bl, then the relative interior of the remaining two vectors is a line segment that misses the origin. If we remove any three of bi, bj, bk, bl, then the relative interior of the remaining vector is empty. Thus Z is a codimension 2 face of P . bj bi α β γ δ bk bl Figure 4.5: The vector arrangement in case 3 of Proposition 4.3.3 It is well known that a codimension 2 face of a polytope is the intersection of exactly two facets. Under our assumptions, we can say more. Proposition 4.3.4. Let (xu,Z) be as above and suppose Z is the intersection of two facets . Then there exists and such that and Z = F1 ∩ F2 xj1 ∈ F1 xj2 ∈ F2 j1 6= j2 1 ≤ j1, j2 ≤ 4. Proof. By Proposition 4.3.3, Z is a codimension 2 face. There are three cases to consider. Case 1. If there are , where , then we are done since is xj1 , xj2 ∈ Z 1 ≤ j1, j2 ≤ 4 Z the intersection of two facets. 31 Case 2. Suppose only one of x1, x2, x3, and x4 are in Z. Without loss of generality, suppose x4 ∈ Z. There are three subcases to consider. (a) If b1 and b3 are linearly dependent, then Z ∪ {x2} is a facet. (b) Suppose the angle between b1 and b3 is less than π. Since Z is a codimension 2 face of P and 0 ∈/ relint(conv(b1, b2, b3)), there must be some j > 4 such that 0 ∈ relint(conv(b1, b2, b3, bj)). Then 0 ∈ relint(conv(b1, b3, bj)) and so Z ∪{x2} is a facet. (c) Suppose the angle between b1 and b3 is greater than π. Since Z is a codimension 2 face and for , there must be some 0 ∈/ relint(conv(bi1 , bi2 )) i1, i2 ∈ {1, 2, 3} such that Then j > 4 0 ∈ relint(conv(b1, b2, b3, bj)) 0 ∈ relint(conv(bj, bi1 , bi2 )) for some i1, i2 ∈ {1, 2, 3}. Thus, the union of Z with the remaining vector is a facet. Case 3. Suppose x1, x2, x3, x4 ∈/ Z. There are three subcases to consider. (a) If b1, b3, and 0 are collinear and b2, b4, and 0 are collinear, then Z ∪ {x1, x3} and Z ∪ {x2, x4} are facets whose intersection is Z. (b) Without loss of generality, suppose that b1, b3, and 0 are collinear and b2 and b4 are linearly independent. If the angle between b2 and b4 is greater than π, then 0 ∈ relint(conv(b2, b3, b4)) and otherwise, 0 ∈ relint(conv(b1, b2, b4)). So either Z ∪ {x1} or Z ∪ {x3} is a facet of P . Since b1, b3, and 0 are collinear, Z ∪ {x2, x4} is a facet of P . (c) Suppose that no two of b1, b2, b3, and b4 lie on a line through the origin. Then there is an i, 1 ≤ i ≤ 4 such that the angle between bi and bi+2 is greater than π. Then 0 ∈ relint(conv(bi, bi+1, bi+2)) and Z ∪ {xi+3} is a facet of P . 32 For the other facet, consider the angle θ between bi+1 and bi+3. If θ > π, then 0 ∈ relint(conv(bi+1, bi+2, bi+3)) and Z ∪ {xi} is a facet. If θ < π, then we have that 0 ∈ relint(conv(bi, bi+1, bi+3)) and Z ∪ {xi+2} is a facet. This concludes case 3. 4.4 A local duality map for codimension 2 toric ideals In this section, we describe an isomorphism for Theorem 4.1.3. Let n X v3 = B1+ + B2+ − ei, v4 = v3 − B1. i=1 v2 = v3 − B2, v1 = v3 − B1 − B2 where is the -th standard basis vector of n.Let 3 . The stan- ei i R f ∈ ExtS(S/IA,S) dard monomials of p r s t form a -basis for 3 . Since the stan- hx , x , x , x i k ExtS(S/IA,S) dard monomials of hxp, xr, xs, xti are indexed by standard pairs, we let fxuxz ∈ 3 where the monomial u z is indexed by the standard pair u . By ExtS(S/IA,S) x x (x ,Z) Theorem 4.3.3 and Theorem 4.3.4, Z is the intersection of 2 facets, F1 and F2, such that there exists and and . We de ne the map xj1 ∈ F1 xj2 ∈ F2 1 ≤ j1 6= j2 ≤ 4 3 n−3 by φ : ExtS(S/IA,S) → Hm (S/IA) u z 1 v v x x 7→ (x j1 ⊕ x j2 ) . (4.2) xuxz 1 Proposition 4.4.1. vj1 vj2 n−3 . xuxz (x ⊕ −x ) ∈ Hm (S/IA) v v v v v v Proof. Since x j1 and x j2 have the same degree, then ∂(x j1 ⊕−x j2 ) = x j1 −x j2 ∈ and so vj1 vj2 is 0 in . Thus vj1 vj2 is in the kernel of IA ∂(x ⊕ −x ) (S/IA)x1···xn x ⊕ −x ∂. 33 v v We now show that x j1 ⊕ −x j2 is not in the image of ∂. To do this, we show that there is no solution 2 to the inequality for . It z ∈ Z Bz ≤ vi − u − z i = 1,..., 4 is enough to show that there is no solution 2 to the inequality z ∈ Z Bz ≤ vi − u − l for i = 1,..., 4. Case 1. i = 1: When i = 1, we are considering the inequality n X Bz ≤ B1+ + B2+ − B1 − B2 − ei. i=1 If z1, z2 > 0, the third inequality fails. If z1 ≤ 0, z2 > 0, the fourth inequality fails. If z1, z2 ≤ 0, the rst inequality fails. If z1 > 0, z2 ≤ 0, then the second inequality fails. Case 2. i = 2: When i = 2, we are considering the inequality n X Bz ≤ B1+ + B2+ − B2 − ei. i=1 If z1, z2 > 0, the fourth inequality fails. If z1 ≤ 0, z2 > 0, the third inequality fails. If z1, z2 ≤ 0, the second inequality fails. If z1 > 0, z2 ≤ 0, then the rst inequality fails. Case 3. i = 3: When i = 3, we are considering the inequality n X Bz ≤ B1+ + B2+ − ei. i=1 If z1, z2 > 0, the rst inequality fails. If z1 ≤ 0, z2 > 0, the second inequality fails. If z1, z2 ≤ 0, the third inequality fails. If z1 > 0, z2 ≤ 0, then the fourth inequality fails. 34 Case 4. i = 4: When i = 4, we are considering the inequality n X Bz ≤ B1+ + B2+ − B1 − ei. i=1 If z1, z2 > 0, the second inequality fails. If z1 ≤ 0, z2 > 0, the rst inequality fails. If z1, z2 ≤ 0, the fourth inequality fails. If z1 > 0, z2 ≤ 0, then the third inequality fails. Proposition 4.4.2. The map φ is well-de ned. Proof. Let u1 z1 u2 z2 3 . Suppose u1 z1 and u2 z2 are indexed x x = x x ∈ ExtS(S/IA,S) x x x x u by the standard pair (x ,Z). Since Z is a codimension 2 face, Z = F1 ∩ F2 for facets F1 and F2. Fix the coordinates corresponding to F1 and F2. By construction, we choose and such that and . Suppose we choose possibly xj1 xj2 1 ≤ j1, j2 ≤ 4 j1 6= j2 dierent indexes for u1 z1 and u2 z2 , say , and 0 , 0 respectively so that x x x x j1 j2 j1 j2 u z 1 v v x 1 x 1 7→ (x j1 ⊕ −x j2 ) xu1 xz1 u z 1 v 0 v 0 x 2 x 2 7→ (x j1 ⊕ −x j2 ). xu2 xz2 Since each component has the same degree, the dierence of the images is zero in local cohomology and so the map is well-de ned. Proposition 4.4.3. Let (xu,Z) be a standard pair of hxp, xr, xs, xti and suppose that 1 u z vj1 vj2 under the map 4.2. Fix a facet of such that , x x 7→ xuxz (x ⊕ −x ) F P Z ⊂ F that is, x a coordinate in the image of . Then xvi properly contains a φ nsupp( xuxz ) codimension 2 face of P and is contained in a facet of P . 35 Proof. Clearly supp(xz) ⊂ F . Since (xu,Z) is a standard pair of hxp, xr, xs, xti, where p , r , and so Z = {x1 . . . , xn}\{xj1 , xj2 , xj3 , xj4 } xj1 ∈ supp(x ) xj2 ∈ supp(x ) on. By Proposition 4.3.3, Z is a codimension 2 face. Let Figure 4.6 represent the the Gale diagram of F . bj2 bj1 bj3 bj4 Figure 4.6: Gale diagram in the proof of Proposition 4.4.3 Fix a 1 ≤ j ≤ 4 such that xj is in a facet that contains Z. Observe that vj nsupp(x ) = {xi : bi and bj are in the same quadrant}. Then if we look at the vector arrangement of the corresponding variables in xvj , the vector arrange- nsupp( xz ) ment Figure 4.7, with possibly additional vectors and no vectors in the j-th quadrant. Since we xed our coordinate in the image of , this means that xvj does φ nsupp( xz ) not contain the codimension 2 face Z. Furthermore, from the vector arrangement, xvj is a subset of one of the facets that contain . nsupp( xz ) Z 36 bj2 bj3 bj4 xv Figure 4.7: Gale diagram of the complement of j nsupp( xu ) u u We now show that dividing by x keeps this inclusion. If xk ∈ supp(x )∩F , then u the inclusion holds. Suppose xk ∈ supp(x ) \ F . We claim that xk gets canceled vj out by x . Now, xk has a dierent sign convention than xj. Note that xk is in the p r s t support of exactly one of x , x , x , and x . The exponent of xk is strictly less than u u the exponent of the monomial that xk appears in since x ∈ supp(x ) and (x ,Z) is a standard pair. But the monomials xp, xr, xs, and xt are made up of the greatest vj vj common divisors of the x 's. Thus, the exponent of xk in x is at least the exponent u u v of xk in x . Thus xk in x cancels with x . Proposition 4.4.4. The map φ in is injective. Proof. Let 3 and n−3 . Then f ∈ ExtS(S/IA,S) f 7→ 0 ∈ Hm (S/IA) X X 1 f = λ · xuxz 7→ λ (xv ⊕ −xv) xuxz If the image of f under the map is 0, then the image is 0 at every coordinate d−2. We now restrict to the coordinate at d−2. Suppose that F ∈ uP F ∈ FP φ(f) F = 0 and X 1 v φ(f) = x j . F xuxz 37 Then 1 vj is contained in and properly contains a codimension 2 face nsupp xuxz x F of P by Theorem 4.4.3. In particular, each summand is not 0. Furthermore, since deg(xv1 ) = ··· = deg(xv4 ), it remains to show that deg(xuxz) are distinct. This follows from the fact that the u z form a -basis for 3 . Thus, if x x k ExtS(S/IA,S) φ(f) = 0, then the coe cients are all 0 and so f = 0. Injectivity of φ follows. Since φ is injective, we have the following consequence by Theorem 4.1.3. Corollary 4.4.5. The map φ in equation 4.2 is also surjective. Thus φ is an iso- morphism. 4.5 Examples In this last section, we illustrate the isomorphism in 4.2. In the rst three ex- amples, NA has one syzygy quadrangle. In the last example, NA has two syzygy quadrangles. Example 4.5.1. Let 2 2 1 1 1 1 −2 3 and . A = B = 0 5 1 11 −1 −4 1 −1 Then 3 1 1 −1 2 4 −1 1 v3 = v4 = v2 = v1 = −1 0 3 4 0 −1 1 0 38 and 3 2 4 3 4 x x x1x 2 x1x x4 x2x xv3 = 1 2 xv4 = 2 xv = 3 xv1 = 3 . x3 x4 x2 x1 Figure 4.8 shows the convex hull of the columns of A and the Gale diagram of B. x1 x2 x3 x4 x1x3 x2 x4 Figure 4.8: conv(A) and the Gale diagram of B in Example 4.5.1 The toric ideal of A in S is 2 2 5 3 2 2 3 4 4 5 IA = hx2x3 − x1x4, x2 − x3x4, x1x2 − x3x4, x1x2 − x3i. The minimal free resolution of S/IA is 2 3 4 3 −x x2 −x −x x x4 x2 1 2 3 3 2 −x 0 −x2 0 2 4 2 −x1 2 2 x3 −x4 x1 x2 −x4 0 x 0 x2 F : S −−−−−→x3 S4 −−−−−−−−−−−−−−−→3 1 S4 −→ S so 1 ∼ 2 2 . The standard pairs of 2 2 are ExtS(S/IA,S) = S/hx1, x2, x3, x4i hx1, x2, x3, x4i 39 and . Then (1, ∅), (x1, ∅), (x1x2, ∅) (x2, ∅) 1 v1 v4 v1 v4 1 7→ x ⊕ x x1 7→ (x ⊕ x ) x1 1 1 v1 v4 v1 v4 x1x2 7→ (x ⊕ x ) x2 7→ (x ⊕ x ). x1x2 x2 Example 4.5.2. Let 1 2 1 1 1 1 1 −1 1 A = 1 0 −1 0 3 and B = −2 −1 . 0 1 −2 0 3 3 −1 −1 −1 Then 2 1 0 −1 0 1 −1 0 v3 = −1 v4 = 1 v2 = 0 v1 = 2 . 2 −1 3 0 −1 0 0 1 Figure 4.9 shows the convex hull of the columns of A and the Gale diagram of B. The minimal free resolution of S/IA is 2 2 3 −x2x3 −x1x4 −x3x5 −x4 −x3x5 −x x1 0 0 x3x5 4 2 x2 x4 −x3x5 −x1 x1x2 F : S −−−−−−→x1 S4 −−−−−−−−−−−−−−−−−−→0 x2 x4 0 S4 −→ S. 40 x5 x1 x2 x2 x1 x4 x3 x5 x4 x3 Figure 4.9: conv(A) and the Gale diagram of B in Example 4.5.2 The standard pairs of hx3x5, x4, x2, x1i are (1, {x3}) and (1, {x5}). The codimension 2 face {x5} is the intersection of facets {x1, x5} and {x2, x5}. For each facet, we only have one xi such that 1 ≤ i ≤ 4 and these variables are distinct. The monomials indexed by the standard pair are of the form e for and so (1, {x5}) x5 e ≥ 0 1 e v1 v2 x5 7→ e (x ⊕ −x ) . x5 The monomials indexed by are of the form f . The codimension 2 face (1, {x3}) x3 {x3} is the intersection of facets {x1, x3} and {x2, x3}. For this example, we choose 1 f v1 v2 x3 7→ f (x ⊕ −x ) . x3 f We could have also sent to 1 v3 v2 instead, along with other choices, but x3 f (x ⊕ −x ) x3 any choice is equal in local cohomology. Example 4.5.3. Let 41 1 1 1 1 1 1 1 1 −1 1 −2 0 0 0 0 1 −1 −2 A = and B = . 0 1 0 1 0 0 1 −1 1 1 2 0 0 1 −2 −1 2 2 Then 1 0 0 −1 0 1 −1 0 −1 0 1 2 v3 = v4 = v2 = v1 = . 0 −1 1 0 −1 1 0 2 3 1 1 −1 Figure 4.10 shows the convex hull of the columns of A and the Gale diagram of B. x3 x6 x2 x2 x1 x6 x1 x4 x5 x4 x5 x3 Figure 4.10: conv(A) and the Gale diagram of B in Example 4.5.3 42 The toric ideal of A in S is 2 2 2 2 2 2 3 3 2 4 IA = hx3x4 − x2x5, x3x4x5 − x1x2x6, x2x3x5 − x1x4x6, x3x5 − x1x6i. The minimal free resolution of S/IA is 2 −x3x5 x1x 0 0 −x x2 6 1 6 x −x x −x x2 −x x2 −x x 4 2 5 1 6 3 5 3 5 2 2 −x4 −x2 x3x4 −x3x5 −x1x6 x F : S −−−−−−→2 S4 −−−−−−−−−−−−−−−−−−−→0 0 x2 x4 S4 −→ S so 3 ∼ 2 . The standard pairs of 2 ExtS(S/IA,S) = S/hx1x6, x3x5, x4, x2i hx1x6, x3x5, x4, x2i are (1, {x5, x6}), (1, {x3, x6}), (1, {x1, x5}), (x6, {x1, x5}), (1, {x3, x1}), and (x6, {x3, x1}). Let e f be a monomial indexed by . The two facets with and x1x5 (1, {x1, x5}) x1 x5 as vertices are the facets x1x4x5 and x1x3x6x5. We make the choice 1 xexf 7→ (xv1 ⊕ xv3 ). 1 5 e f x1x5 If e f is a monomial indexed by . Then x1x5 x6 (x6, {x1, x5}) 1 x3xf x 7→ (xv1 ⊕ xv3 ). 1 5 6 e f x1x5 x6 Example 4.5.4. This examples demonstrates the case when IA has two syzygy quadrangles. Let 5 1 1 1 1 1 −3 2 and A = B = . 1 6 0 13 −3 −2 1 −1 43 Figure 4.11 shows the convex hull of the columns of A and the Gale diagram of B. x1 x2 x3 x4 x3x1 x2 x4 Figure 4.11: conv(A) and the Gale diagram of B in Example 4.5.4 The two syzygy quadrangles of IA are the unit square Q and the quadrangle Q · s−1 as shown in Figure 4.12. Figure 4.12: The two syzygy quadrangles of IA. The matrix B · s−1 is 4 1 −5 2 −1 B · s = . −1 −2 2 −1 44 The toric ideal of A in S is 2 2 3 3 5 5 4 2 6 5 7 3 3 IA = hx1x2 − x3x4, x2x3 − x1x4, x2x3 − x1x4, x1 − x2x3, x2 − x1x3x4i. The minimal free resolution of S/IA is 2 2 x2 x3 0 −x2 2 0 x4 −x3 0 x1 0 F : S2 −−−−−−−−→x4 x1 S6 −−−−−→ S5 −−−−−→ S. The copy of S in homological degree 3 corresponding the unit square is the map given by the second column and is twisted by 9 . The copy of in homological degree 3 ( 31 ) S that corresponds to the rhomboid is the map given by the rst column and is twisted by 9 . The twist can be checked by computing −1 −1 . Then ( 43 ) (B · s )1+ + (B · s )2+ 1 ∼ 9 2 9 2 2 . The standard pairs ExtS(S/IA,S) = S ( 43 ) /hx1, x2, x3, x4i ⊕ S ( 31 ) /hx1, x2, x3, x4i of 2 are and . The standard pairs of 2 2 are hx1, x2, x3, x4i (1, ∅) (x2, ∅) hx1, x2, x3, x4i , , , and . The map corresponding the the unit square (1, ∅) (x2, ∅) (x2x3, ∅) (x3, ∅) is done as in Example 4.5.1. Then 4 0 3 −1 1 6 −1 4 v3 = v4 = v2 = v1 = . −1 0 1 2 1 −1 2 0 45 The copy of S corresponding to the rhomboid, 1 v3 v4 v3 v4 1 7→ x ⊕ x x2 7→ (x ⊕ x ). x2 46 5. ASSOCIATED PRIME IDEALS OF LOCAL COHOMOLOGY MODULES OVER STANLEY-REISNER RINGS While local cohomology modules are not nitely generated in general, they may possess other niteness properties that enable us to gain an understanding of their structure. One such property is the number of associated prime ideals of the lo- cal cohomology modules. Let R be a k-algebra where k is a eld of arbitrary characteristic. Huneke asked if the number of associated prime ideals of a local cohomology module i is nite [6]. Huneke's question has been answered in HI (R) the a rmative for smooth Z-algebras [3], Gorenstein rings of nite F-representation type [16], among other rings. There do exists examples by Katzman [8], Singh [14], and Singh and Swanson [15] where local cohomology modules do not have nitely many associated prime ideals. One such example due to Singh [14] is if we let and . Then 3 has in nitely R = Z[u, v, w, x, y, z]/hux + vy + wzi I = hx, y, ziR HI (R) many associated prime ideals. Madsen, Wheeler, and I answered Huneke's question in the a rmative when R is a Stanley-Reisner ring over a eld of arbitrary characteristic whose associated simplicial complex is a T -space [2]. Our approach was inspired by Lyubeznik's characteristic- free approach in [9]. 5.1 Simplicial complexes and Stanley-Reisner rings An (abstract) simplicial complex on the vertex set V = {x1, . . . , xn} is a collection of subsets of V that is closed under inclusion, that is, if σ ∈ ∆ and τ ⊂ σ, then τ ∈ ∆. We assume {xi} ∈ ∆ for all i. The elements of ∆ are called faces. The maximal faces of ∆ under inclusion are called facets. The dimension of a face σ ∈ ∆ is dim σ = |σ| − 1. The f-vector of ∆ is the vector (f−1, f0, . . . , fdim ∆−1) where fi is 47 the number of i-dimensional faces of ∆. The dimension of ∆ is the maximum of the dimensions of the faces of ∆. 0 0 If V ⊂ V is a subset of the vertices of ∆, de ne ∆V 0 := {σ ⊂ V : σ ∈ ∆}. The link of a face σ ∈ ∆ is and link∆(σ) = {τ ∈ ∆ : σ ∪ τ ∈ ∆ σ ∩ τ = ∅}. The star of a vertex x ∈ V is st(x) = {σ ∈ ∆ : σ ∪ {x} ∈ ∆} and the core of the vertex set V of ∆ is core∆(V ) = {v ∈ V : st(x) 6= V }. De ne core ∆ := ∆core V . A face σ ∈ ∆ can be separated from a face τ ∈ ∆ if there exists a facet containing σ but not containing τ. De nition 5.1.1. A simplicial complex ∆ is a T -space if for every face σ ∈ ∆, if τ 6⊂ σ, then σ can be separated from τ. The following equivalent criterion for a T -space is found in [19]. Proposition 5.1.2. Let ∆ be a simplicial complex on vertex set V . Then 1. ∆ is a T -space if and only if σ may be separated from {v} for any face σ ∈ ∆ and vertex v ∈ V \ σ. 2. Given a face σ ∈ ∆ and a vertex v ∈ V \ σ, then σ can be separated from {v} if and only if σ ∩ core V can be separated from {v} in core ∆. 48 The T -space property on a simplicial complex should be thought of as a simplicial analogue to the separation axioms in topology. Figure 5.1 and Figure 5.2 show examples of simplicial complexes that are not T -spaces. Figure 5.3 and Figure 5.4 show examples of T -spaces. c d a b Figure 5.1: A simplicial complex that is not a T -space c d a e b Figure 5.2: A pure simplicial complex that is not a T -space c d a b Figure 5.3: A pure T -space 49 b e d f a c Figure 5.4: A non-pure T -space Let be the polynomial ring in variables over a eld and let S = k[x1, . . . , xn] n k ∆ be a simplicial complex on the vertex set V = {x1, . . . , xn}. The Stanley-Reisner ideal, or face ideal, of ∆ in S is the ideal generated by the non-faces of ∆ I∆ := hxi1 ··· xit : {xi1 , . . . , xit } ∈/ ∆i ⊂ S. The Stanley-Reisner ring, or face ring, of ∆ over k is the quotient ring S k[∆] := . I∆ The Krull dimension of k[∆] is dim k[∆] = dim ∆ + 1. Proposition 5.1.3. [4, Theorem 5.1.7] The Hilbert function of a Stanley-Reisner ring with -vector is k[∆] f f = (f0, f1, . . . , fd−1) 1, t = 0 H(k[∆], t) = . Pd−1 d−1 i=0 fi i , t > 0 50 The Hilbert series of k[∆] under the ne grading is X Y λi F (k[∆], λ) = . 1 − λi σ∈∆ xi∈σ 5.2 Dierential operators on Stanley-Reisner rings Let R be a commutative k-algebra. We de ne the ring of k-linear dierential operators on as follows. Set viewed as a subring of the -algebra R D0(R; k) = R k by the multiplication map . For each , let Homk(R,R) r 7→ (s 7→ rs) i ≥ 0 for each Di+1(R; k) = {P ∈ Homk(R,R):[P, r] ∈ Di(R; k) r ∈ R}, where the bracket [P, r] = P · r − r · P is the commutator and the product is com- position. The -subalgebra of , k Homk(R,R) [ D(R; k) = Di(R; , k), i≥0 is the ring of -linear dierential operators on R. The elements of are the k Di(R; k) dierential operators of order i on R. This gives R the structure of a left D(R; k)- module. When we reference R with its D(R; k)-module structure, we call R a D- module. Example 5.2.1. Let k be a eld of characteristic 0 and R = k[x] be the polynomial ring over k. Then D(R; k) is ∂ ∂ D(R; k) = khx, ∂x :[ ∂x , x] = 1i, where ∂ is the usual partial dierential operator for . ∂x x 51 If the characteristic of k is p > 0, then D(R; k) is generated by R and all the divided powers 1 ∂t for . t! ∂xt t > 0 Example 5.2.2. More generally, let be a eld of characteristic 0 and k R = k[x1, . . . , xn] be the n-variate polynomial ring over k. Then D(R; k) is ∂ ∂ ∂ ∂ ∂ D(R; ) = hx1, . . . , xn, ,..., :[xi, xj] = [ , ] = 0, [ , xj] = δiji. k k ∂x1 ∂xn ∂xi ∂xj ∂xi If the characteristic of k is p > 0, then D(R; k) is generated by R and all the divided 1 ∂t powers t for i = 1, . . . , n and t > 0. t! ∂xi For the remainder, we use the following notation for partial dierential operators. Notation 5.2.3. Let t ∂t be the -th divided power for . If ∂i = t t xi a = (a1, . . . , an) ∈ ∂xi n a a1 a2 an ∂f , then ∂ = ∂ ∂ ··· ∂ . We write if the partial dierential operator for xi N 1 2 n ∂xi is acting of f. When ∂i and f are being multiplied in as elements in a D-module, we write ∂i · f or ∂f. For the remainder of this section, we examine the D-module structure of Stanley- Reisner rings. Theorem 5.2.4. [4] Let k[∆] be a Stanley-Reisner ring over a eld k. Then the minimal prime ideals of I∆ are in bijection with the facets of ∆. The minimal prime of corresponds to the face of comple- P = hxi1 , . . . , xit i I∆ mentary vertices {xj : i 6= i1, . . . , it}. We use the above characterization of minimal primes of along with the following theorem of Traves [18] to determine . I∆ D(k[∆]; k) Theorem 5.2.5 (Theorem 3.5). [18] Given a monomial ring R = k[x]/J with no nonzero nilpotent elements, an element of the Weyl algebra xa∂b is in D(R) if and only if for each minimal prime P of R, we have either xa ∈ P or xb ∈/ P . 52 In particular, D(R; k) is generated as a k-algebra by {xa∂b : xa ∈ P or xb ∈/ P for each minimal prime P of R}, and these are a free basis for D(R; k) as a left k-module. Tripp [19, Theorem 5.7] showed that for a Stanley-Reisner ring k[∆] over a eld of characteristic 0, D(k[∆]; k) is left Noetherian if and only if core ∆ is a T -space. Theorem 5.2.6. Let R = k[∆] be a Stanley-Reisner ring where core ∆ is a T -space. Then t D = D(R; k) = Rhxi∂i : 1 ≤ i ≤ n, t ≥ 0i. Proof. Theorem 5.2.5 gives that D is generated as a k-module by xa∂b such that for every minimal prime P of k[∆], either xa ∈ P or xb ∈/ P . The operators x∂t satisfy this condition so t . Rhxi∂i : 1 ≤ i ≤ n, t ≥ 0i ⊆ D For the reverse inclusion, suppose that xa∂b ∈ D. We assume supp(xa) ∈ ∆ a since otherwise x ∈ I∆. The facets of ∆ correspond to the minimal prime ideals of a a b I∆. Choose a facet σ containing supp(x ). Then x ∈/ Pσ and by [18] supp(x ) ⊂ σ. Hence, supp(xb) ∈ ∆. Furthermore, supp(xa) cannot be separated from supp(xb) and so supp(xb) ⊂ supp(xa) since ∆ is a T -space. It follows that for all i = 1, . . . , n we have and so bi divides a b. ai ≥ bi xi∂i x ∂ Following the convention in [18], a D-submodule is called a D-stable ideal. Theorem 5.2.7. Let k[∆] be a Stanley-Reisner ring such that core ∆ is a T -space. Then 1. An ideal J ⊂ k[∆] is D-stable if and only if it is a squarefree monomial ideal. 2. The length of as a -module is . k[∆] D λD(k[∆]) = |∆| 53 Proof. 1. By [18, Lemma 4.4], every D-stable ideal is a square free monomial ideal. For the converse, let J ⊂ k[∆] be a squarefree monomial ideal. It su ces to show that is stable under t. Since is a monomial ideal, it is enough J xi∂i J to show stability for monomials. Let a . If , then t . x ∈ J ai < t xi∂i = 0 ∈ J t a ∂ x ai a−b Otherwise, xi t = x , where b = (0, . . . , t − 1,..., 0) with t − 1 in t!∂xi t position i. Since ai ≥ t, |a − b| = |a| 2. We induct on the number of facets of ∆. Let ∆0 be a obtained by deleting one facet of ∆. Then k[∆0] is a D-submodule of k[∆]. Moreover, any D-submodule of I∆0 must also be a squarefree monomial ideal and so the submodule corre- sponds to a simplicial complex between ∆0 and ∆. Since ∆0 is obtained by deleting one facet of ∆, this is not possible. Hence λD(I∆0 ) = 1. By induction, 0 . Therefore . λD(k[∆]/I∆0 ) = |∆ | = |∆| − 1 λD(k[∆]) = 1 + (|∆| − 1) = |∆| Lemma 5.2.8. The partial dierential operators on satisfy the fol- ∂i k[x1, . . . , xn] lowing relation t 2 t t−1 (5.1) xi∂i xi = x ∂i + tx∂i where we let 0 . ∂i = 1 Proof. We induct on t. For t = 1, multiplying the Leibniz rule on the left by x gives x∂x = x2∂ + x. 54 Assume equation 5.1 for t − 1. Then ∂tx = ∂(∂t−1x) = ∂(x∂t−1 + (t − 1)∂t−2) = ∂x∂t−1 + (t − 1)∂t−1 = (x∂ + 1)∂t−1 + (t − 1)∂t−1 = x∂t + t∂t−1. Multiplying on the left by x gives the desired result. Lemma 5.2.9. For any f, g ∈ R and j ≥ 0, t g 1 X 1 ∂if j g x∂t = x∂t(g) − x∂t−i f j f j f j ∂xi f j i=1 with x∂0 = x. Proof. Multiplying the product rule on the left by x gives t X ∂f x∂t(fg) = x∂t−i(g). ∂x i=0 Rewriting the above equation as an operator on g gives t X ∂if x∂tf = x∂t−i. ∂xi i=0 55 Thus t X ∂if x∂tf − x∂t−i = ∂0(f) · x∂t ∂xi i=1 = f · x∂t. Replacing with j and then multiplying on the left by 1 gives f f f j t 1 X 1 ∂if j x∂t = x∂tf j − x∂t−i. f j f j ∂xi i=1 Applying the above operator to g gives the result. f j ∈ Rf We remark that the proof of Lemma 5.2.9 describes how Rf inherits a D-module structure from R. Furthermore, Lemma 5.2.9 holds if we replace g with an element from D-module so we emphasize that Mf inherits a D-module structure from M. Proposition 5.2.10. Let be a -rational maximal ideal m = (x1 − c1, . . . , xn − cn) k of k[∆], that is, the natural map k ,→ k[∆]/m is a bijection. 1.A -basis for D/Dm is given by {x ··· x ∂t1 ··· ∂tl : {x , . . . , x } ∈ ∆}. k a1 al a1 al a1 al 2. For any w ∈ D/Dm that is not in the eld k, there exists f ∈ m such that fw = 1. Proof. 1. Note that {xa∂t : supp(xa)∪supp(xt) ∈ ∆} is a k vector space basis for D. We now determine the relations among these generators in D/Dm. Denote x ··· x ∂t1 ··· ∂tl such that {x , . . . , x } ∈ ∆ by x∂t. Suppose a ∈ n is a a1 al a1 al a1 al N a−ei t vector with ai ≥ 1 for some xed i and consider x x∂ (xi − ci) where ei is the i-th standard basis vector with a 1 in the i-th coordinate and 0 in all other coordinates. Note that these elements are a k-vector space basis for Dm. 56 If ti ≥ 2, then by Lemma 5.2.8 a−ei t a−ei t a−ei t x x∂ (xi − ci) = x (x∂ xi) − cix x∂ a t a−ei a−ei t = x x∂ + tix − cix x∂ . ai−ei t Since x x∂ (xi − ci) ∈ Dm, we have the following congruence in D/Dm a t ai−ei t−ei a−ei t x x∂ ≡ −tix x∂ + cix x∂ . (5.2) If ti = 0, then a−ei t a t a−ei t x x∂ (xi − ci) = x x∂ − cix x∂ , which implies we have the congruence a t a−ei t x x∂ (xi − ci) ≡ cix x∂ . (5.3) Lastly, when ti = 1, Lemma 5.2.8 gives a−ei t a−ei t a−ei t x x∂ (xi − ci) = x (x∂ xi) − cix x∂ a t a t−ei a−ei t = x x∂ + x x∂ − cix x∂ . Thus, in D/Dm, a t a t−ei a−ei t x x∂ ≡ −x x∂ + cix x∂ (5.4) a−ei t−ei a−ei t ≡ −cix x∂ + cix x∂ where the second congruence follows from equation 5.3 applied to xax∂t−ei . Observe that in each of equations 5.2, 5.3, and 5.4, the exponent vectors a − ei 57 occurring on the right side of the equations are smaller coordinatewise than the exponent vectors a occurring on the left side of the equations. We then use these congruences to reduce any element of D/Dm to terms with exponent vector 0. Hence we reduce to linear combinations of elements in {x ··· x ∂t1 ··· ∂tl : a1 al a1 al and so this set is a basis for as a -vector space. {xa1 , . . . , xal } ∈ ∆} D/Dm k 2. Applying the congruences in 5.2, 5.3, and 5.4 in the case a = ei and rearranging gives −t x∂t−ei , t > 1 i i t t−ei (xi − ci)x∂ ≡ −cix∂ , t = 1 0, ti = 0. a t In particular, we see that (x−c) with c = (c1, . . . , cn) annihilates x∂ if ai > ti for any i while (x − c)tx∂t ∈ k∗. s Write P tj where ∗. Pick any such that is the maximum w = j=1 bjx∂ bj ∈ k l tl among all 0 and let . Let 0 a. By our choice of , for any tjs a = tl f = (x − c) l 0 tj j 6= l, there is some index i for which ai > ti, so that f annihilates x∂ . 0 0 tl 0 tl Thus f w = bl(f x∂ ), and f x∂ is a unit by our observations above. Put f = (f 0w)−1f 0 to get fw = 1. Proposition 5.2.11. Let m ⊂ k[∆] be a maximal ideal, let M be a D-module, and let z ∈ M be an element such that ann (z) = m. The set {x ··· x ∂t1 ··· ∂tl · z : R a1 al a1 al is linearly independent over {xa1 , . . . , xal } ∈ ∆} k Proof. Let ˜ denote the algebraic closure of and let ˜ ˜ ˜ , ˜, k k R = k⊗kk[∆] = k[∆] m˜ = mR ˜ ˜ ˜ ˜ , and ˜ ˜ . Then ˜ is a ˜-module and ˜ . D = k ⊗k D = D(R; k) M = k ⊗k M M D M ⊂ M It su ces to show that {x ··· x ∂t1 ··· ∂tl · z : {x , . . . , x } ∈ ∆} is linearly a1 al a1 al a1 al 58 independent in ˜ over ˜. Since ˜ is separable, ˜ is reduced. Thus there M k k/k k ⊗k (R/m) are maximal ideals such that Ts . Since ˜ is algebraically closed, m1,..., ms m = i=1 mi k each is -rational. By the Chinese remainder theorem, mi k s ! s ˜ ˜ ∼ ˜ ˜ ∼ ˜ M ˜ ∼ M ˜ ˜ (5.5) D/Dm˜ = D ⊗R˜ (R/m˜ ) = D ⊗R˜ (R/mi) = (D/Dmi). i=1 i=1 Since , there is a -module map m˜ = annR˜(z) D φ :D/˜ D˜m˜ → M˜ w 7→ w · z. We claim that φ is injective. Indeed, suppose there is some w ∈ D/˜ D˜m˜ such ˜ ˜ that w · z = 0 but w 6= 0. For 1 ≤ i ≤ s, let wi be the image of w in D/Dmi. By Proposition 5.2.10, there is some fi ∈ mi such that fiwi = 1. It follows that there is some such that ˜ for all and at least one . Then under f ∈ R fwi ∈ k i fwi 6= 0 the correspondence above in 5.5, the tuple (fw1, . . . , fws) corresponds to a nonzero element ˜ . Then and so , a fw ∈ R/m˜ (fw) · z = f(w · z) = 0 fw ∈ annR/˜ m˜ (z) = 0 contradiction. Thus φ is injective. By Proposition 5.2.10, the set {x ··· x ∂t1 ··· ∂tl : {x , . . . , x } ∈ ∆} is lin- a1 al a1 al a1 al ˜ ˜ ˜ ˜ early independent in D/Dm˜ since it is independent in each D/Dmi. Therefore, its image {x ··· x ∂t1 ··· ∂tl · z : {x , . . . , x } ∈ ∆} is linearly independent in M˜ . a1 al a1 al a1 al 5.3 Associated primes of local cohomology modules of Stanley-Reisner rings Let R be a homomorphic image of the polynomial ring S over a eld k. The ring of dierential operators has a natural ltration by total degree D(R; k) B = B0 ⊂ 59 B1 ⊂ · · · , where n a b X Bt = k ·{x ∂ : (aj + bj) ≤ t} ∩ D(R; k), j=1 called the Bernstein ltration. For the remainder, B will denote the Bernstein ltra- tion on D(R; k).A k- ltration on a D-module M is an ascending chain of k-vector spaces such that S and .A -module is F0 ⊂ F1 ⊂ · · · i Fi = M BiFj ⊂ Fi+j D M holonomic if it has a - ltration with dim R where is a k F0 ⊂ F1 ⊂ · · · dimk Fi ≤ Ci C constant independent of i. Proposition 5.3.1. Let M be a D-module and let z ∈ M such that P = annR(z) is a prime ideal in . Then for all , Pd . R d dimk(Bdz) ≥ i=0 H(R, i) Proof. Let h = height P , K be the eld of fractions of R/P , and let Q ⊂ k[x] be a prime ideal such that P = QR. After permuting the variables of S, we may assume that are algebraically independent over in , and is nite over xh+1, . . . , xn k K K . Set , . L = k(xh+1, . . . , xn) R = L ⊗T R M = L ⊗T M Note that σ = {xh+1, . . . , xn} is a face of ∆, and R = L[Γ] where Γ = link∆ σ. The ideal P R = annR(z) is a maximal ideal of R so by Proposition 5.2.11, the set {x ··· x ∂t1 ··· ∂tl : {x , . . . , x } ∈ Γ} a1 al a1 al a1 al is linearly independent in M. Hence the set Y := {(xal+1 ··· xan )x ··· x ∂t1 ··· ∂tl : {x , . . . , x } ∈ Γ} l+1 n a1 al a1 al a1 al is linearly independent in M. Let Yd = Y ∩ Bdz. Then Yd ⊆ Bdz and since the elements of are linearly independent, we have the inequality . Yd dimk Bdz ≥ |Yd| 60 Under the correspondence t t+1, the set is in bijection with the set of xi∂i 7→ xi Yd monomials not in the Stanley-Reisner ideal I∆ of degree at most d. Hence |Yd| = Pd . j=0 H(R, j) Theorem 5.3.2. Every holonomic D-module M has nite length in the category of D-modules and l(M) < C · n! for some constant C. Proof. Let F = F0 ⊂ F1 ⊂ · · · ⊂ Fl = M be a D-module ltration of M and let be a - ltration on . Then there is a constant such that d G k M C dimk Gi ≤ Ci for all . For each , induces a ltration (s) on where (s) i 1 ≤ s ≤ l G G Fs/Fs−1 Gj = (Gj ∩ Fs)/(Gi ∩ Fs−1). Choose a prime Ps ∈ AssR(Fs/Fs−1). Then there exists zs ∈ Fs/Fs−1 such that . Assume that (s). Then for all , s. By Ps = annR(zs) zs ∈ Gjs i ≥ js Bi−js zs ⊂ Gi Proposition 5.3.1, (s) Pi−js . Therefore dimk Gi ≥ j=0 H(R, j) l l i−js r X (s) X X Ci ≥ dimk Gi = dimk Gi ≥ H(R,J). s=1 s=1 j=0 Since the Hilbert function eventually equals a polynomial of degree r − 1 and the leading coe cient is at least 1 , the function Pi−js is eventually a poly- (r−1)! j=0 H(R, j) nomial of degree with leading coe cient at least 1 . Thus l and so . r r! C ≥ r! l ≤ Cr! Finally we get λD(M) ≤ Cr! ≤ ∞. Before continuing, we recall Faulhaber's formula on the sum of p-th powers of the rst m positive integers. Proposition 5.3.3 (Faulhaber's formula). The sum of pth powers of the rst m positive integers is m p X 1 X p + 1 ap = (−1)i B mp+1−i p + 1 i i a=1 i=0 61 where Bi is the i-th Bernoulli number. Theorem 5.3.4. Let be a Stanley-Reisner ring whose associ- R = k[x1, . . . , xn]/I∆ ated simplicial complex is a T -space. Then R is a holonomic D-module. Proof. Let r = dim R and F = F0 ⊂ F1 ⊂ · · · be a ltration on R where Ft consists of the elements of R with degree at most t under the standard grading. Then F is a k- ltration. We show that there is a constant such that for all , d. By C i dimk Fi ≤ Ci Proposition5.1.3, t t X X dimk Ft = H(R, i) = 1 + H(R, i) i=0 i=1 t r−1 ! X X i + j = 1 + (−1)r−1−je . r−1−j j i=0 j=0 Let e be the maximum multiplicity e = max0≤j≤r−1 er−1−j. Then t r−1 X X e dim F ≤ 1 + (i + j)(i + j − 1) ··· (i + 1). k t j! i=0 j=0 For each l, the a-th coe cient of the polynomial (i+j)(i+j −1) ··· (i+1) is the sum of all products consisting of factors chosen among . There are j choices a 1, . . . , j a with the largest such product equal to j! so we get j(j − 1) ··· (j − a + 1) = (j−a)! t r−1 j 2 X X e X j dim F ≤ 1 + a! ia k t j! a i=1 j=0 a=0 t r−1 j 2 X X e X a!j = 1 + e ia j! j! a i=1 j=0 a=0 Since , the binomial coe cients satisfy j r−1 for any and so we get j ≤ r − 1 a ≤ a a 62 the inequality a!j2 j! (r − 1)! 1 (r − 1)! = ≤ · ≤ . j! a (j − a)!2 (r − 1 − a)! (j − 1)! (r − 1 − a)! We now obtain a bound independent of j as follows. t r−1 j X X X (r − 1)! dim F ≤ 1 + e ia k t (r − 1 − a)! i=1 j=0 a=0 t r−1 r−1 X X X (r − 1)! ≤ 1 + e ia (r − 1 − a)! i=1 j=0 a=0 t j X X (r − 1)! = 1 + e r ia (r − 1 − a)! i=1 a=0 r−1 t X (r − 1)! X = 1 + er ia (r − 1 − a)! a=0 i=1 r−1 a ! X (r − 1)! X (−1)m a + 1 = 1 + er B ta+1−m . (r − 1 − a)! a + 1 m m a=0 m=0 The last line is a degree r polynomial in t. To get an explicit bound on all the coe cients, it remains to eliminate an index. Let B = max0≤m≤r−1 |Bm|. For any a, (r − 1)! r − 1 r − 1 r − 1 (5.6) = a! ≤ a! r−1 ≤ (r − 1)! r−1 . (r − 1 − a)! a b 2 c b 2 c By inequality 5.6, r−1 r − 1 X r dim F ≤ 1 + er r(r − 1)! Btr−m. k t b r−1 c m 2 m=0 We let C be the largest coe cient that appears r − 1 r C = eBr r! r−1 r b 2 c b 2 c 63 and conclude that R is holonomic. Theorem 5.3.5. Let R = k[∆] be a Stanley-Reisner ring whose associated simplicial complex is a T -space. Let D = D(R; k) be the ring of k-linear dierential operators on R and suppose that M is a D-module. Then for any f ∈ R, if M is holonomic, then so is Mf . In particular, Rf is a holonomic D-module. Proof. Let d be the degree of the polynomial f, r = dim R, and let B : B0 ⊂ B1 ⊂ · · · be the Bernstein ltration. Since M is holonomic, there is a k- ltration on M, such that r for some constant . De ne the ltration F : F0 ⊂ F1 ··· dimk Fi ≤ Ci C 0 0 0 by F : F0 ⊂ F1 ⊂ · · · u F 0 = · : u ∈ F . i k f i i(d+1) If F 0 is a k- ltration, then 0 r dimk(Fi ) ≤ dimk Fi(d+1) ≤ C(i(d + 1)) . 0 r Setting C = C(d + 1) gives that Mf is holonomic. It remains to show that 0 is a - ltration. First note that S 0 . For F k i Fi ⊆ Mf the reverse inclusion, choose u where for some . If , f w ∈ Mf u ∈ Fi i i ≤ w(d + 1) then and hence, u 0 by de nition. On the other hand, if u ∈ Fi ⊆ Fw(d+1) f w ∈ Mw j j i > w(d + 1) the put j = i − w(d + 1). Since f ∈ Bjd, it follows that f u ∈ Fjd+i. Rewriting jd + i = jd + (j + w(d + 1)) = (j + w)(d + 1) gives u f j u 0 S 0 and so S 0 . f w = f j+w ∈ Fj+w ⊆ i Fi i Fi = Ff We now show that for all , 0 0 . Let u 0 with . By i, j BiFj ⊆ Fi+j f j ∈ Fj u ∈ Fj(d+1) Lemma 5.2.6, we show that for , we have the containment t 0 0 . We t ≥ 0 xi∂i Fj ⊆ Ft+1+j induct on t. For the base case, we have xi∂i = xi. Since xif ∈ Bd+1, it follows that 64 . Then u xfu 0 as desired. (xif)(u) ∈ Fj(d+1)+(d+1) x f j = f j+1 ∈ Fj+1 Assume for that t−l u 0 . By Lemma 5.2.9, it su ces to 1 ≤ l ≤ t (x∂ · f j ∈ F(t−s+1)+j show that each of 1 1 ∂lf j u x∂t · u and · x∂t−l · f j f j ∂xl f j are in 0 . Since , t , and , we have F(t+1)+j f ∈ Bd x∂ ∈ Bt+1 u ∈ Fj(d+1) t+1 t f · x∂ ∈ Bd(t+1)+t+1 = B(t+1)(d+1) and hence t+1 t f · x∂ (u) ∈ F(t+1)(d+1)+j(d+1) = F(t+1+j)(d+1). Thus 1 f t+1x∂t(u) x∂t(u) = ∈ F 0 . f j f t+1+j t+1+j For the other term, there exists an element us ∈ F(t−s+1+j)(d+1) such that u u x∂t−s = s f j f t−s+1+j by the induction hypothesis. We claim by induction on that j−s divides ∂sf j . But this is just the usual j f ∂xs power rule in . When , we have ∂s(1) . Assume the claim as our D(S; k) j = 0 ∂xs = 0 s j−1 inductive hypothesis. Write ∂ f as a quotient vs,j−1 . Using the usual product rule ∂xs f s−j+1 65 for higher order derivatives gives t ∂t(f j−1 · f) X ∂sf j−1 ∂t−sf = · ∂xt ∂xs ∂xt−s s=0 t X vs,j−1 vt−s,1 = · f s−j+1 f (t−s)−1 s=0 t X vs,j−1vt−s,1 = . f s−j+1+t−s+1 s=0 We conclude that ∂tf j divides j−t and since was arbitrary, we replace it with . ∂xt f t s Thus we have 1 ∂sf j u v u · ∂t−s = s,j s . (5.7) f j ∂xs f j f t+j The polynomial ∂sf j has degree so the polynomial has degree ∂xs dj − s vs,j dj − s − d(j − s) = ds − s. Hence vs,jus ∈ Fds−s+(t−s+1+j)(d+1) ⊂ F(t+1+j)(d+1) because (t−s+1+j)(d+1)+s(d−1) < (t−s+1+j)(d+1)+s(d+1) = (t+1+j)(d+1). Thus vs,j us 0 and by equation 5.7, we are done. f j ∈ Ft+j 66 REFERENCES [1] Roberto Barrera. Quasidegrees: a package to compute quasidegrees. http://www.math.tamu.edu/ rbarrera/software/. [2] Roberto Barrera, Jerey Madsen, and Ashley K. Wheeler. Finiteness of associated primes of local cohomology modules over stanley-reisner rings. arXiv:1609.05366, 2016. [3] Bhargav Bhatt, Manuel Blickle, Gennady Lyubeznik, Anurag K Singh, and Wenliang Zhang. Local cohomology modules of a smooth Z-algebra have nitely many associated primes. Inventiones mathematicae, 197(3):509519, 2014. [4] Winfried Bruns and H. Jürgen Herzog. Cohen-Macaulay rings. Cambridge University Press, 1998. [5] David Eisenbud. Commutative algebra : with a view toward algebraic geometry. Graduate texts in mathematics. Springer, New York, Berlin, Heildelberg, 1995. [6] David Eisenbud and Craig Huneke. Free Resolutions in Commutative Algebra and Algebraic Geometry. Research Notes in Mathematics. Taylor & Francis, 1992. [7] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a soft- ware system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/. [8] Mordechai Katzman. An example of an in nite set of associated primes of a local cohomology module. Journal of Algebra, 252(1):161166, 2002. [9] Gennady Lyubeznik. A characteristic-free proof of a basic result on d-modules. Journal of Pure and Applied Algebra, 215(8):20192023, 2011. 67 [10] Laura Felicia Matusevich, Ezra Miller, and Uli Walther. Homological methods for hypergeometric families. J. Am. Math. Soc., 18(4):919941, 2005. [11] Ezra Miller and Bernd Sturmfels. Combinatorial commutative algebra. Graduate texts in mathematics. Springer, New York, 2005. [12] Irena Peeva and Bernd Sturmfels. Syzygies of codimension 2 lattice ideals. Math. Z, 229:163194, 1998. [13] Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama. Gröbner deforma- tions of hypergeometric dierential equations. Algorithms and Computation in Mathematics. 6. Berlin: Springer., 2000. [14] Anurag K Singh. p-torsion elements in local cohomology modules. Mathematical Research Letters, 7:165176, 2000. [15] Anurag K Singh and Irena Swanson. Associated primes of local cohomology modules and of frobenius powers. International Mathematics Research Notices, 2004(33):17031733, 2004. [16] Shunsuke Takagi and Ryo Takahashi. D-modules over rings with nite f- representation type. Mathematical research letters, 15(2):563581, 2008. [17] Rekha R Thomas. Lectures in geometric combinatorics, volume 33. American Mathematical Soc., 2006. [18] William N Traves. Dierential operators on monomial rings. Journal of Pure and Applied Algebra, 136(2):183197, 1999. [19] J Tripp. Dierential operators on stanley-reisner rings. Transactions of the American Mathematical Society, 349(6):25072523, 1997. 68 APPENDIX A MACAULAY2 SCRIPTS FOR QUASIDEGREES This appendix contains summaries and the source code for the main functions of the Macaulay2 package Quasidegrees. The main function of the package Quasidegrees is the method quasidegrees. The input is a nitely generated module over the polynomial ring that can be presented as a monomial matrix. The output is a list Q that represents the quasidegree set of M. The list Q consists of pairs (u,F ) where u ∈ Zd and F is a list of vectors in Zd. The pair represents P . If the user inputs an ideal instead (u,F ) u + b∈F C · b I ⊂ R of a module, quasidegrees is executed on the module R/I. quasidegrees=method() quasidegrees(Module) := (M) -> ( R := ring M; P := presentation M; if not isMonomialMatrix P then error "Expected module to be presented by a matrix with monomial entries"; if not isPositivelyGraded M then error "Module is not positively graded"; if not isHomogeneous M then error "Module is not homogeneous with respect to ambient grading"; E := entries P; D := degrees target P; -- We make the following list of pairs S. -- The first entry is a degree twist in the presentation of M. -- The second entry are the standard pairs of the corresponding row. S := apply(#D, i -> ( {vector(D_i), standardPairs monomialIdeal E_i} ) ); -- Next we make a list T representing the quasidegree set of M as follows. -- The variables in S get assigned their degrees and then 69 -- shifted by the corresponding degree twist in the -- presentation of M. T := flatten(apply(S, s-> ( apply((s_1), w -> ( if w_0==1 then ( if w_1 =={} then ( {s_0,{}} ) else {s_0, apply(w_1, x -> vector degree x)} ) else{s_0 + (vector(degree(w_0))), apply(w_1, x -> vector degree x)} ) ) ) ) ); toList set T ) quasidegrees(Ideal) := (I) -> ( R := ring I; M := R^1/I; quasidegrees M ) The other main function of Quasidegrees, and the motivation for writing the package Quasidegrees, is the function quasidegreesLocalCohomology, abbreviated QLC. The input for QLC is an integer i and a nitely generated module M over a polynomial ring that is presented by a matrix with monomial entries. QLC returns a list Q representing the quasidegree set of the i-th local cohomology module of M supported at , i . Local duality is used to compute the local cohomology m Hm(M) module. As in quasidegrees, the list Q consists of pairs (u,F ) where u ∈ Zd and is a list of vectors in d. The pair represents P . If the user F Z (u,F ) u + b∈F C · b inputs an ideal I ⊂ R instead of a module, QLC is executed on the module R/I. 70 quasidegreesLocalCohomology = method() quasidegreesLocalCohomology(ZZ, Module) := (i,M) -> ( R := ring M; n := numgens R; v := gens R; e := vector sum apply(v,x -> degree x); -- use Local Duality N := Ext^(n-i)(M,R); P := presentation N; if not isMonomialMatrix P then error "Expected module to be presented by a matrix with monomial entries"; if not isPositivelyGraded N then error "Module is not positively graded"; if not isHomogeneous N then error "Module is not homogeneous with respect to ambient grading"; E := entries P; D := degrees target P; -- We make the following list of pairs S. -- The first entry is a degree twist in the presentation of M. -- The second entry is a list of the standard pairs of the monomial ideal -- generated by the entries of the corresponding row. S := apply(#D, i -> ( {vector(-D_i), standardPairs monomialIdeal E_i} ) ); -- We next make a list T representing the quasidegree set of M. -- The variables in S are assigned their degrees and then -- shifted by the corresponding degree twist in the -- presentation of M. T := flatten(apply(S, s-> ( apply((s_1), w -> ( if w_0==1 then ( if w_1 =={} then ( {s_0,{}} ) else {s_0, apply(w_1, x -> vector degree x)} ) else{s_0 - (vector(degree(w_0))), apply(w_1, x -> vector degree x)} ) 71 ) ) ) ); Q := toList set T; apply( #Q, j -> {((Q_j)_0)-e,(Q_j)_1}) ) quasidegreesLocalCohomology(ZZ, Ideal) := (i,I) -> ( R := ring I; M := R^1/I; quasidegreesLocalCohomology(i,M) ) 72